436 recent mathematical tablesFile Function and Accuracy Range and Interval

13 meridional arc (.001 meters) 0(1')90°A (10D, 8S), B (10D, 8S) 0(1')90°

14 C(15D,7S),Z? (12D,4S),£ (17D.5S), F(16D,4S) 0(1')90°15 N, R (both .001 meters) 0(1')90°16 military grid coordinates from U.S.C. & G.S.S.P.

no. 59X, Y (both .1 yard) Lat., Long, of U. S. 5' intersections

17 latitude transformation tables geometric-isometric: isometric-geodetic for followingspheroids: Clarke 1866, 1880, 1860; Hayford,Bessel, Everest 0(1')90°geodetic-authalic: authalic-geodetic for Clarke 1866 0(1')90°

U. S. NAVAL OBSERVATORY

File Function and Accuracy Range and Interval

1 sin x (7D), cos x (7D), tan * (7D) 0(0?01)90°2 1/x (6D) 1 (.001)4

4(.01)103 log^x^D) 0(.001)14 hav"1 x (about 0?001) 0 to 25°5 sin x (8D), cos x (8D) 0(0?01)90°

Special tables for astronomical work.

vega aircraft corporationFile Function and Accuracy Range and Interval

1 log x (8D), a log x (8D), 1/(a log x) 10000(1)100000W. J. E.

RECENT MATHEMATICAL TABLES

211[A].—W. Vriesendorp, The Calculating Dictionary, Utrecht, L. E. Bosch& Zoon, printer, for the author, 1937. [1003 p.], 13.5 X 26.7 cm. 17.5Dutch florins.This Dictionary contains the multiplications up to 999 X 999, except multiples of ten.

There are two pages of examples showing applications of the volume in obtaining theproducts of larger numbers. On each page headed 2,3, . . ., 999 are given 900 products.The Dutch edition has the title De Rekendiclionnaire Tabellen bijeengebrackl ten dienstevan handel, industrie, administratie en onderwijs.

There is no reference to any other work offering the same tabular information, andreadily available in 1937, such as: A. L. Crelle, Calculating Tables, Berlin, 1930 which hasgone through many editions, since, the first two-volume edition in 1820, including one byan insurance company in Japan (1913).

212[D].—M. J. Buerger & Gilbert E. Klein, "Correction of X-ray-intensities for Lorentz and polarization factors," Journal of AppliedPhysics, v. 16, July, 1945, p. 414-418. 20 X 26.6 cm.Four-place tables are given for the following functions: 2 sin 29/(1 + cos2 28) and

2/(1 + cos2 28), for the argument sin 8 = 0(.001).999.

recent mathematical tables 437

213[D, M].—Leon Beskin, "General circular ring analysis," Aircraft Engi-neering, v. 17, May, 1945, p. 127-132. 24.5 X 31 cm.There are tables of d>(0) = [l/2»rp(l + cos 9) - i(3 sin 9)], '(0) = [l/2x][- 9 sin 6

- i cos 9 + 1], 4>"ifi) = [1/2tt][- I sin 9 - 9 cos 9], tf>"'(9) = [l/2x][- i(3 cos 9) + 9 sin 9],*(9) = *(«) + *"(») = [l/2x][9 - 2 sin 9], *'(9) = *'(9) + 4>'"W = [l/2x][l - 2 cos 9],^"(9) =

438 recent mathematical tables

16 to 22D. Tables IIB, C, and D tabulate the same functions over the ranges x = .1(.1).9;.01(.01).09; .001(.001).009, the first function to 24 or 25D and the second to 18D.

Tables IIIA, B, C give values of e1 over the following ranges: x = [0(.1)9.9; 18D],x = [0(.001).099; 24D], x = [0(.001).999, 17D].

Tables HID, E, and F tabulate e", where y is successively x-10-6, x-10~9, and x-10~12over the range x = [0(1)999; 20D].

Table IV provides to 18D the first ten multiples of M and l/M, where M = log e.The errors of this volume are discussed elsewhere in this issue, MTE 68.In the year following the publication of the work under review, the extensive tables

of the NYMTP made their appearance. These cover the following range: For e?,x = [0(.0001)1; 18D], x = [1(.0001)2.S(.001)5; 1SD], and x = [5(.01)10; 12D]; for er*,x = [0(.0001)2.5; 18D]. It is thus seen that, to 10 decimal places, the Holtappel tables,even now, extend the range of values tabulated, especially for the exponent of descendingargument.

H. T. D.

215[E].—NYMTP (A. N. Lowan, technical director), Tables of the Expo-nential Function ex, New York, 1939, xviii, 535 p. 21 X 27 cm. Repro-duced by a photo-offset process. Sold by the U. S. Bureau of Standards,Washington, D. C. $2.00.This volume provides accurate tables of the function e* for small intervals of the

argument. The frequent use of the exponential function in a wide variety of computationalwork makes these tables extremely valuable as they give a complete tabulation of thefunction over a large range of values. An account of the care with which the tables weredesigned and checked is set forth in the Introduction.

The first four tables give the values of the ascending exponential from 0 to 10 withvarying intervals and number of decimals. The ranges are as follows: x = [0(.0001)1; 18D],[1(.0001)2.5(.001)5; 15D], and [5(.01)10; 12D].

The values of the descending exponential are given for the range — x = [0(.0001)2.S;18D].

Besides these main tables, there are three tables to cover more adequately the intervalabout zero, and certain integral values of x, nameiy: ± x «= [0(.000001).0001; 18D],[1(1)100; 19S], and [1(1)9-10-*; 18D], p = 7(1)10.

It is unfortunate that the tables for e~~* were not extended beyond x = 2.5. The valuesof the function for arguments larger than 2.5 can, of course, be calculated by using thesupplementary integer table. To be sure there are Newman's tables1 for x = [0(.001)15.349;12D], [15.350(.002)17.3(.005)27.635; 14D], and [.1(.1)37; 18D]. However, it would havebeen a decided advantage to have had the complete tables collected into one volume.

Evelyn FixUniv. California,Berkeley

1 F. W. Newman, "Table of the descending exponential function to twelve or fourteenplaces of decimals," Cambridge Phil. So., Trans., v. 13, 1883, p. 145-241.

216[E, L].—National Defense Research Committee, Tables for Solu-tions of the Wave Equation for Rectangular and Circular Boundaries havingFinite Impedance, prepared by A. N. Lowan for the NYMTP, & P. M.Morse, H. Feshbach & E. Haurwitz for the M.I.T. Underwater SoundLaboratory. Report dated June 1945. Printed from manuscript, ii, 52leaves, and 7 plates on only one side of each leaf, by the photo-offsetprocess. These tables are available only to certain Government agenciesand activities.For rectangular boundaries the wave equation can be separated into differential equa-

RECENT MATHEMATICAL TABLES 439

tions of the form

(1) dty/dx1 + *V = 0.The boundaries for the x coordinate can be placed at x = 0, and x = a. The boundarycondition at x = a will be denned by the equation

(2) ^ = zx{d\p/dx), (x = a),

where z = re** is proportional to the impedance of the boundary at x = a. The solution ofthe problem depends on the nature of the boundary conditions at x = 0.

Case I. Boundary condition at x = 0 is 4> = 0.

In this case the solution used is

(3) \// = sinh (ikx) =

where w = ika. The boundary condition at xrise to the equation

(4) (l/w) tanh w

where

(5) w = R - il = rße-rt.

The values of R and I, for each of four branches are given, Table I, p. 7-26, as functions ofr and

440 recent mathematical tables

yieldsJi(iw) = ± iJ

recent mathematical tables 441Form Insert Delete

x2 + 42y2 157 159x2 + 66 f 71 77x2 + &6y* 87 89x2 + 89y2 345 354x2 + 10iy 309, 317, 333 305, 321x2 - 38y> 21, 131 23, 129x2 - 62y 107, 141 103, 145

There is also an error in (ii), which was correct in edition (c), namely:

p = 13, T. I, argument 12, for -, read 6.

Then in the rest of the volume under review are 8 papers (p. 173-282), a demonstrationof a theorem of Chebyshev based on a fragment found among his papers, edited by Markov(p. 283-284), and Commentary by Delone (p. 285-309). The paper on Quadratic Forms (p.208-228) contains a table (p. 220-222) of quadratic forms ± (x2 - 2y*), ■ ■ -, ± (x2 - 33y*),with limits of x and y, and linear forms of N. This table had appeared earlier in J. d. Math.,s. 1, v. 16, 1851, p. 273-274, and in Sochineniia-Oeuvres, v. 1, 1899, p. 88-89.

R. C. A.1 There are now the following six editions of the Theory of Congruences:

(a) Teorita Sravnenii (Theory of Congruences), Diss. St. Petersburg, St. Petersburg, 1849,ix, iii, 281 p.

(b) Second ed., St. Petersburg, 1879, viii, iii, iii, 223, 26 p.(c) Third ed., St. Petersburg, 1901, xvi, 279 p.(d) Fourth ed., Moscow-Leningrad, 1944, reviewed above.(e) Theorie der Congruenzen, German transl. by H. Schapira, Berlin, 1889, xviii, 314, 32 p.(f) Teoria delle Congruenze, Italian transl., with additions and a note, by I. Massarini,

Rome, 1895, xvi, 295 p.

218[H].—Henry A. Nogrady, A New Method for the Solution of CubicEquations, Ann Arbor, Edwards Bros., 1936, iv, 22, ii, xxx p. 13.5 X 21cm. Lithoprinted. $1.25.Given a cubic equation with numerical coefficients,

(1) ax3 + bx* + cx + d = 0,

the substitution x = y — b/(3a) transforms equation (1) into

(2) y> + py + q = 0,where p = (3ac - 2>2)/(3a2), q = 2b3/(27a3) - 6c/(3a) + d/a. Applying the transformationy = qz/p to equation (2), we get

(3) z3 + nz + n = 0,

where n = p3/q2. If Zi is one of the three roots of (3) n = — zis/(zi + 1). Hence, while Ziincreases from — 1.5 to — 1, re decreases from — 6.75 to — =« ; while zi increases from — 1to 0, n decreases from + Z\ > — 1.5, n may take on any value from + °° to — =».

In the thirty-page table, for each z1 (except — 1) in this range, at interval .001, thecorresponding value of n is given to 6D (except for a few values to a larger number of decimalplaces). Thus, having reduced a given cubic equation to the form (3), an approximatethree-place value of Zi, corresponding to the resulting n, can be read off from the table.Then the other roots of (3), real or imaginary, are found by the formulae

z2 = *z,{- 1 + [(2l - 3)/(a, + 1)1»}, z3 = iz,j- 1 - [(z, - 3)/(z, + 1)]»}.

The author shows that if zi + d is the true value of zi a six-place value for zi may be foundby one or more applications of the formula Zi + d = (2zi» — re)/(3zl2 + n).

A tiny 6-page Pocket Tables for Cubics. A Systematic Method for Algebraic Treatmentof Cubic Equations, 1933, by David Katz, patent attorney, was reviewed in Scripta Mathe-matics, v. 2, 1934, p. 379. The cubic is reduced to the form z3 + z = D.

The numerical solution of cubic equations, with tables, is also set forth by F. Emde,

442 recent mathematical tables

Tables of Elementary Functions, 1940, p. 38-47, see MTAC, p. 384-385. The standardform here is y3 + 2 = 3py. If only one root is real the roots are yi = y' + iy" = si",y»-y- »/' = »-'.yi = - 2y' = - 2s cos o-. For 3p = - 9.9(.l) + 1(.05)2(.02)2.8(.01)3,Emde gives y', y", s, to 4S or 5S, and a to 4D or 5D. When all the roots are real, the rootsy%, 3% ys are given to 4S or 5S, for 3p = 3(.01)3.2(.02)4(.05)5(.1)10(.5)15. See also Jahnke& Emde, Tables of Functions . . ., 1943 and 1945 (MTAC, p. 386), Addenda, p. 20-30.

Tables for the solution of the trinomial equations xm+n±exm±f'=0, are given inS. Gundelfinger, Tafeln zur Berechnung der reellen Wurzeln sämtlicher trinomischenGleichungen - ■ -, Leipzig, 1897.

R. C. Ä.

219[J, L,M].—I. M. Ryzhik, Tablitsy Integralov, Summ, Riadov i Proiz-vedenii [Tables of Integrals, Sums, Series and Products], Leningrad,OGIZ, 1943. 400 p. 14.6 X 21.5 cm. 25 roubles, paper bound.Edition of 3000 copies.Since the author felt that no single Soviet or foreign work presented an adequate

collection of formulae in integrals, sums, series, and products, for the research mathe-matician and engineer, the present work with over 5000 formulae was compiled to fill theneed. While the work is primarily for such workers, in order to extend the circle of peoplewho might profit from its use, a concise body of indispensable information, supplementingthe formulae, is given near the end of the work (p. 339f).

The main sources for the choice of formulae were, (a) for indefinite and elliptic integrals,W. Läska, Sammlung von Formeln der reinen und angewandten Mathematik, Braunschweig,1888-1894; (b) for definite integrals Bierens de Haan, Tables d'Integrales Definies, Amster-dam, 1858-64, and Nouvelles Tables d'Integrales Definies, Leyden, 1867; (c) for sums, series,and products, E. P. Adams, Smithsonian Mathematical Formulae and Tables of EllipticFunctions, Washington, 1922.

The author tells us that the classification of the material presented great difficulties.The arrangement of the indefinite integrals is essentially that given by G. H. Hardy in hisIntegration of Functions of a Single Variable (Cambridge, second ed., 1928); the authorfound it desirable, however, to assemble the material on elliptic integrals and functionsaccording to his own arrangement, in Part II. The classification of definite integrals is, withminor modifications, that of Bierens de Haan.

Great attention was paid to special functions, particularly elliptic, cylindrical, spherical.The book contains many formulae pertaining to these functions. Among others are givenalso formulae of a function introduced by the author namely (p. 296):

six, y) = rfe + y — l)/T(x)T(y) = lim s(x, n)s(y, n)/s(x + y — 1, n)n-*oo

= fr (x + k)(y + k)

The introduction of this function made it possible to simplify and to generalize a number offormulae.

An outline of the contents is as follows:Part I: Indefinite Integrals (p. 1-80)

I. General formulae; 2. Fundamental integrals; 3. Rational functions; 4. Irrationalfunctions; 5. Trigonometric functions; 6. Exponential functions; 7. Logarithmicfunctions; 8. Inverse trigonometric functions; 9. Hyperbolic functions.

Part II: Elliptic Integrals and Functions (p. 81-111)1. Fundamental formulae; 2. Elliptic integrals; 3. Integrals of elliptic functions;4. Elliptic integrals and Weierstrass functions.

Part III: Definite Integrals (p. 112-239)1. General formulae; 2. Integrals of elementary functions; 3. Special functions;4. Multiple integrals.

recent mathematical tables 443

Part IV: Sums, Series, Products (p. 240-338)1. Numerical series and products; 2. Fundamental series and products (elementaryfunctions); 3. Special functions.The theoretical section on Applications (p. 339-387) has the following subheadings:1. Integration in finite form; 2. Approximate methods of integration; 3. Multiple

integrals; 4. Convergence of series and products; 5. Classification of series and products;6. Transformation of series; 7. Connection between series and products; 8. Methods ofsummation of power series.

Then follow four numerical tables (p. 388-391) of frequent occurrence: (1) (2n — 1)!!/(2»)!1 = s(n + 1, «+l)/22\ n = 1(1)15; coefficients in the expansion of (1 + *)*;(2) (2n - l)!!/[(2»)!!(2» + 1)] = s(n + 1, n+ l)/[22»(2« + 1)], n = 1(1)15;(3) (2n - l)!!/(2« + 2)!! = s(n + 2, n + 2)/22»+2(2ra + 1), n = 1(1)14;(4) (2n - l)!!/[(2» + 2)!!(2rc + 3)] = s(n + 1, n + l)/(22»+1s(2» + 2, 3)], « = 1(1)14.All of these numerical values are given as fractions and to 10 decimal places; up to n = 10,(1) and (3), in decimal forms, are given in P. Barlow, New Mathematical Tables, London,1814, p. 256; up to n = 15 the exact decimal values of (1) are given in J. H. Lambert,Supplementa Tabularum, Lisbon, 1798, p. 198; also German ed., 1770, p. 210.

Of the 47 titles in the Bibliography (p. 392-393), no dates are given for 41. Most ofthe titles are well-known works, and it is of interest that only Russian editions are men-tioned for works of Courant, La Vallee-Poussin, Euler, Goursat, Granville, Scarborough,Whittaker & Robinson, and Whittaker & Watson. Gray & Matthews' Treatise on BesselFunctions, L. B. W. Jolley's Summation of Series, B. O. Peirce's A Short Table of Integrals, I.Todhunter's An Elementary Treatise on Laplace's Functions, Lame's Functions and Bessel'sFunctions, and Watson's A Treatise on the Theory of Bessel Functions, are naturally includedin the list. The only items unknown to the reviewer were two Russian works:

Nina K. Bary, Theory of Series,I. M. Ryzhik, Special Functions.Miss Bary has been a professor of mathematics at the Univ. of Moscow since 1932.

A somewhat detailed index of the formulae fills p. 394-400.The work is undoubtedly one of considerable value for any mathematician to have

at hand.R. C. A.

220[L].—K. E. Bisshopp, "Lateral bending of symmetrically loaded conicaldiscs," Quarterly Appl. Math., v. 2, Oct. 1944, p. 214-217. 17.6 X 25.4cm. See RMT 202.The calculation of the deflection coefficients and stress coefficients depends upon the

hypergeometric functions

Gi{x) = Gi(l - x) = F{ia, ib; 1; (1 - 2x)2|,Gi{x) = - G2(l -*)-(!- 2x)F{\a + J, ib + }; 3/2; (1 - 2x)'\.

The tables give Gt(*), G\(x), G2(x), (?'»(*)• for x = [0(.01).5; 6 or 7SJ.There are also tables, with 6 or 7S, for the deflection and stress coefficients which are

computed by numerical integration from the functions Gi(x), Gi{x), and two auxiliaryfunctions Hi(x), Hi(x). The H-functions are called "subtracting off" functions and arechosen so that the differences, G„(x) — Hn{x) are bounded uniformly throughout the intervalof existence of Gn(x). Some intelligence is used in the choice of these functions.

H. B.

221[L].—N. Karsmenkov in Herbert Buchholz, "Die konfluente hyper-geometrische Funktion mit besonderer Berücksichtigung ihrer Bedeutungfür die Integration der Wellengleichung in den Koordinaten eines Rota-tionsparaboloi'des," Z. angew. Math. Mech., v. 23, 1943, p. 106-108, 117.21.6 X 27.8 cm.

444 recent mathematical tables

This is a continuation of a former paper, p.. 47-58, in which definitions are given ofwave-functions m{^(if), w^(it) suitable for the treatment of problems connected with aparaboloid of revolution. The relation between m${iS) and Whittaker's confluent hyper-geometric function Mh.m(z) is

The tables on p. 106 give from 4 to 7S for

P/r)%£(4)for t = - 3 (.5)3 and f = 1(1)6. Those on p. 107 give from 5 to 7S for

(2A)W«(if)A)ffor the same ranges. On p. 106 there is also a short table of the smallest root ti of the equa-tion

m^m = 0to 5D for f = 0(1)6 and f = 4.80966 when n = ± 0. A diagram gives a perspectiverepresentation of A^.•r.o(^^)/(*f), for — 3 < t < 2, 0 < f < 6. On p. 107 there is a perspec-tive representation of the function

a?for t = 0 to 3, f = 0 to 6. On p. 108 there is a short table of the first three zeros r'i, t\, t'3of m'f^(i(). The computations were made by N. Karsmenkov who also drew the diagrams.Among the asymptotic formulae that are given, mention may be made of one for

when f < 0 and t is large and positive, a series for 2\|Tmf| in descending powers of jo„,where Jo(jon) = 0, asymptotic formulae for M„, jp(z) and W„, jp(z) as z -*• », v —*■ andas p—+ «, some of these being due to Whittaker and Erdelyi. Orthogonal relations aregiven, one of which is claimed to be new as the arguments'of the functions w!^'n(ff) areimaginary. It is thought that the series, asymptotic formulae and orthogonal functionsgiven in the paper represent a decisive step forward towards the solution of the practicallyimportant problems relating to the paraboloid of revolution. A good bibliography is given(p. 117) but in the light of some information about what is being done in this country thelist is far from complete.

H. B.

222[L].—Murlan S. Corrington & William Miehle, "Tables of Besselfunctions J„(x) for large arguments," /. Math. Physics, M.I.T., v. 24,Feb. 1945, p. 30-50. 17.4 X 25.5 cm.Five-place tables of Jn(ms) are here given for n = 0(1)10, 5 = 1(1)20, and (a)

m = 1(1)10; (b) m = x(x)5x. Also for m = 1 and x, n = 0(1)10, s - 21(1)40. These arethe tables to which earlier reference was made in their unpublished form, MTAC, p. 285.

The values of Jn(s), n — 1(1)10, j = 1(1)24 were obtained by rounding off the 18-placevalues of Meissel, published in A. Gray & G. B. Mathews, A ^Treatise on Bessel Functions,London, 1895; second ed. 1922. For higher values of the argument the values were allcomputed from the asymptotic formula. The auxiliary functions An{x), Ba(x), Ai(x), Bi(x),and their tabulations (see MTAC, p. 282), were used when possible. Tabular values werecomputed to 7D or 8D, and later rounded off to 5D. "Where there was any reasonabledoubt as to whether to add or drop a half, the function was recalculated more accurately."As a further check comparison was made with the following known tables: (a) H. Nagaoka,Tokyo, College of Science, J., v. 4, 1891, where there is a table of /o(rex), n = [1(1)50; 6D];a 5-place abridgement is in K. Hayashi, Fünfstellige Funktionentafeln . . ., Berlin, 1930;no errors were found. See MTAC, p. 299. (b) Values of J„(20), /„(30), /„(40), J„(50),

recent mathematical tables 445

/„(100),m = 0,1, are given in K. Hayashi, Tafeln der Besseischen, Theta-, Kugel-, und andererFunktionen, Berlin, 1930; the Statement of the paper under review in this connection isincorrect. See MTAC, p. 291. (c) L. Steiner, Math, naturw. Berichte aus Ungarn, v. 11,1894, which has a table of Ji(x), x = [20.1 (.1)31 (.2)41; 6D]; on comparing the values forintegral argument no discrepancies were found. See MTAC, p. 305.

"The mathematical theory of frequency-modulated radio broadcasting shows that thesideband amplitudes of a carrier modulated with sinusoidal variations in frequency areproportional to the Bessel functions of the first kind J„{m), where n is the sideband numberand m is the modulation index. Since m equals the maximum deviation in frequency, D,divided by the audio frequency n, it is evident that at the lower audio frequencies the ratioD/n can become quite large. This means that in order to determine the sideband amplitudesit is necessary to use tables of Bessel functions for large values of the argument m."

R. C. A.

223[L].—H. R. F. Carsten & Miss N. W. McKerrow, "The tabulation ofsome Bessel functions Kr(x) and K/(x) of fractional order," Phil. Mag.,s. 7, v. 35, Dec. 1944, p. 816-818. 17.1 X 25.5 cm."Under certain conditions, the temperature field within a cylindrical rod, subjected to

a sudden change in temperature, may be developed in terms of modified Bessel functions ofthe second kind, of order n ± \, and their derivatives. As values of such functions do notyet appear to have been published, it has been found desirable to prepare tables of thesefor a range of the variables." Here are tables of K\(x), K3n{x), K3ii(x), Kut{x), K3n(x),- K[„(x), - K'ilt(x), x = [.1 (.1)5,6(2) 10; mostly 5S]. Kk(x) = (w/2x)^ = \; K3ll(x)= X[l + l/x]; Ksn(x) = X[l + (3/x) + 3 fx1]. Then Ki/i(x), K3u{x) were computed byLagrange's interpolation formulae from values of K0, K\, K2 tabulated in G. N.Watson, Theory of Bessel Functions, second ed., 1944, p. 737, and from previously calculatedKm(x), K3ll(x) K'yi(x) = - l/(4x)KlH(x) - K3li(x), K3li(x) = - 3/{4,x)K3li(x)- Kmix).

224[L].—K. Franz & T. Vellat, "Der Einfluss von Trägern auf dasRauschen hinter Amplitudenbegrenzern und linearen Gleichrichtern,"Elek. Nach. Tech., v. 20, 1943, p. 185, 188-189. 21.5 X 28 cm.The calculations in this paper depend on confluent hypergeometric functions, and a

15

table gives C„(x), for n = [1(1)13, 15; 4S], with graphs, and 2 C„(x), to 3S, and x = 0, .1,n = l

.15, .25, .4, .6, 1, 1.5, 2.5, 4, 6, 10, wherexi»-u-2V2(n - k - ^tFlin - k - i;» - 2k; - x*)

Cn(x) = (l/4,r) 2*=oo>» (*!)(»-*)![(»- 2k - 1)!?

H. B.

225[L].—C. Truesdell, "On a function which occurs in the theory of thestructure of polymers," Annals Math., s. 2, v. 46, Jan. 1945, p. 150.17.5 X 25.3 cm.Appell's integral is defined as

x t'~ldt(x, -$), 4>{x, (x, — £), .45 ^ x ^ .80;for d>(x, i), .65 ^ x Si .97; for d>(x, 3/2), .75 ^ x ^ .999. In these ranges the tables are cor-rect to within ±.001.

446 recent mathematical tables

226[L].—[F. Vandrey], Great Britain, Department of Scientific Researchand Experiment, Admiralty Computing Service, Tables of LegendreFunctions Qn(x), London, February, 1945, 2 p. text and 2 p. tables. No.SRE/ACS65. 21.5 X 34.3 cm. Mimeographed. This edition of thesetables is available only to certain Government agencies and activities.

In MTAC, p. 190, appears a review of tables, 1940, computed by F. Vandrey, of theLegendre Function of the second kind, for n = 0(1)7, and for x = [0(.01)1; 5D]. The presentpublication is a reprint of these tables with the following corrections:

Q2(x), x = .82, for -0.64134, read -0.64164Qi(x),x = .51, for -0.31312, read -0.31316Qt(x), x = .99, for +0.03725, read -0.03725Q,(x), x = .99, for -0.07590, read -0.21288

"The differences suggest that the last figure given [in the tables] is probably not more than1 or 2 units in error."

There is a table of 2Qn(x), n = 1(1)8, forx = [0(.01)1; 4D] in H. Tallqvist, Grundernaaf Teorin for Sferiska Funktioner, jämte Användningar inom Fysiken, Helsingfors, 1905,p. 401-407.

227[L, M].—A. H. A. Hogg, "Equilibrium of a thin slab on an elasticfoundation of finite depth," Phil. Mag., s. 7, v. 35, Apr. 1944, p. 270-275.17 X 25.4 cm.There are tables, to 4D, for the definite integrals

(l/2xx) f°J1(mx)F(m)mdmr b = 0(.1)1,

(&2/2jrx2) f™J0(mx)F(m)dm, b = 0(.1)1,2,

recent mathematical tables 447

229[M].—Georges Goudet & Miss A. M. Gratzmuller, "Divergence parTerTet de la charge d'espace d'un faisceau electronique cylindrique nonaccelere," J. d. Physique et le Radium, s. 8, v. 5, July 1944, p. 144-147.21.5 X 28 cm.

/(*) = 4 f Tsinh-' (-, * =^ - sinh-' (- * =) 1 du,

d>a(x) = Ea/Rp = ${f(x) - /(a - *)], Ha) = a J0 a(x), a = .5(.5)5,7,10,20, x/a= [0(.1).5; 3SJ; and of +(a), a = [0(.1)2.S, 2.8(.2)3.2, 3.5, 3.6(.4)4.4, 4.5, 4.8(.2)5.2(.4)-6.8(.2)7.2, 8(1)20; 2-4S]. There are graphs of /(*), 0 < * < 2.5; of 4>«(x) for the 13 valuesof a, 0 < x/a < .5; and of ^(a).

230[U].—Akademha Nauk. S.S.S.R., Leningrad, Matematicheskil Institutimeni V. A. Steklova, Tablitsy dim opredeleniia linii polozheniiä korabliapo radiopelengu. [Tables for the determination of the line of position of aship by radio bearings.] Moscow and Leningrad, Academy of Sciences,1944, 137 p. + errata slip. 12.7 X 20 cm. 10 roubles. 1000 copies in theedition.These tables permit the rapid determination of lines of position from radio bearings of

radio stations within 270 nautical miles of the observer, in latitudes 80° S to 80° N. Theywere prepared by the Mathematical Institute of the Academy of Sciences, on the requestof the Nauchno-ispytatel'nyl Hidrografichesko-shturmanskn Institut [Scientific-experi-mental Pilot Institute]. The basic ideas are credited to L. A. Liusternik, D. A. Vasil'kovand I. fa. Akushskii. The computations were carried out by calculating machine at theInstitute; it is stated that bearings calculated by the method will be correct to within 15'.Since bearings determined by radio-direction-finding apparatus available for general usebefore the war were usually given to the nearest degree, and often were in error by severaldegrees, especially when the sunrise or sunset line lay between the radio station and theobserver, the tables would appear to provide ample accuracy.

The book is quite small and convenient to use. Only a single multiplication is requiredand that could be carried out by slide rule. The paper used in the volume reminds one thatit is a war-time product.

In the use of the tables, one starts with an assumed position, Po, which is chosen insuch a way that AX, Ad>, the differences in longitude and latitude between this position andthat of the radio station are integral numbers of degrees. One then seeks in the table thetabular latitude nearest that of the radio station, and with ax, Ad> as additional arguments,takes out Ac, the computed bearing; i, a correcting angle; and k, a differential coefficientto be used in allowing for the difference between the observed and computed bearings,A — Ae. The line of position can then be laid off in such a way that it makes an angle,T = Ae -4- i ± 90° with the local meridian, and that it lies at a perpendicular distance,k(A — Ac) nautical miles from P0. Rules are given for deciding the side of Po on which theline is to be drawn.

The trigonometric relations used in preparing the tables are:

cot Ac = tan d>o cos 4>c esc AX — sin c tan AX,

k = sin »'/(tan Ac tan d>c),

where Xo, 4>o are respectively the assumed longitude and latitude of the observer, Xe, 4>eare the longitude and latitude of the radio station and AX = Xo — Xc.

It is interesting to note that the tabulated values of latitude are not evenly spaced,nor are they integral degrees; for example, 0° 00', 3° 25', 6° 50', 10° 15', 12° 49', 15° 13',17° 16', ■ • •, 60° 05', 60° 30', 60° 55', 61° 10', 61° 35', • • •, 79° 33', 79° 40', 79° 47', 79° 54'.

448 mathematical tables-errata

The values of ax range from 1° to 28°; and Ad, from - 5° to + 5° for d> = 0, ax = 1°;to - 1° to + 1° for d, = 79°36', ax = 28°.

C. H. SmileyBrown Univ.

231[U].—Great Britain, H. M. Nautical Almanac Office, AstronomicalNavigation Tables, Volume Q, Latitudes N 70°-N 79°, Air Publication1618. London, H. M. Stationery Office, 1945, iv, 341 p. 16.5 X 24.8 cm.These tables are available only to certain Government agencies andactivities.This is the fifteenth and last volume in the series of no. 1618 which has had restricted

circulation in this country by the Hydrographie Office, under the number H. O. 218. Wehave already reviewed the earlier volumes, MTA C, p. 82f, each one covering five degreesof latitude, v. A, 0°-4° (no volume lettered I or 0) to v. P, 65°-69°, the fourteenth. Thepresent volume covering 10° is naturally the largest, and is applicable between latitudes69° 30' and 79° 30' north for specially selected stars, and both north and south for the restof the volume.

R. C. A.

232[U].—Samuel Herrick, "The air almanac refraction tables," U. S.Naval Inst., Proc, v. 70, Sept. 1944, p. 1140-1141. 17 X 25.5 cm.In this note Herrick shows how, by graphical representation, the advantages of critical

tables can be had in the case of double-entry tables. As illustrations, he chose the tables fortotal refraction and refraction adjustment as given in the American Air Almanac. Withheight above sea level in feet, and observed altitude in degrees as the ordinate and abscissarespectively, one reads the total refraction (or refraction adjustment) directly from theappropriate graph. Herrick constructed his graphs from data in L. J. Comrie, Hughes'Tables for Sea and Air Navigation (see MTAC, p. Ill) and notes that there is a slight dis-crepancy between the figures given by Comrie and those presented by the American AirAlmanac.

C. H. Smiley

MATHEMATICAL TABLES—ERRATAReferences have been made to Errata in RMT 216 (N.D.R.C.), 217

(Chebyshev), 222 (Corrington & Miehle), 226 (Vandrey); N 43 (Euler,Legendre, Newman, Powell); QR 18 (Hayashi, Roman).

67. James Burgess, "On the definite integral (2/?rI)./uie_i2